I'll have to look that one up, the variation used in the paper only contains height (h), distance to the listener (r), and D/S spacing (d):

P = I(signal) * V(pol) / (2 Pi d h sqrt(c r f) )

hi,

With sqrt(f) in the denominator, this tells you that with constant current fed to an ESL line source the SPL will fall -3dB/octave as frequency increases. Note also that sqrt(r) is in the denominator so response falls -3dB for each doubling of distance from the line source; a well known phenomenon.

But, we usually think in terms of applied voltage since our amplifiers and step-up transformers are good approximations of voltages sources. Take a look at equation (5) from the paper where the Walker type equation is converted to Vsignal instead of Isignal:

P=V(signal)*V(pol)*C*sqrt(f) / (d*h*sqrt(c*r))

Now sqrt(f) is in the numerator. So, with constant voltage fed to an ESL line the response will rise +3dB/octave with increasing frequency. This implicitly takes care of the dipole roll-off for low frequencies and directivity factor for high frequencies. This rise is exactly what the R-C transmission line is use to equalize out.

For derivition and discussion of point source and line source Walker equations, see sections 3.3.2 and 3.3.11 of Baxandall's ESL chapter in the book "Loudspeaker and Headphone Handbook".

I think kavermei already addressed this question, but I thought I'd add a few comments. Yes, 4 sections is all that is needed to achieve flat on-axis response from 300hz - 20kHz. But, like you said, if you want to use a minimum section width of 1/2" and only 4 sections, you won't have much panel area, so your ESL won't be very loud.

When designing a panel, I'd suggest starting out by:
1) defining the largest panel(height & width) you can live with. This will give you the highest sensitivity and maxSPL.
2) Then, pick your LF breakpoint. With area already defined in 1), this also sets the max SPL for your design.
3) Figure how many sections you need to reach beyond 20kHz.
4) If your sections are already 1/2" or whatever you deem "small enough" then you are done. Otherwise, increase the number of sections until you get there. Your high frequency extension will be much higher then needed, but the leakage inductance of your transformer will most likely cut of the HF response at not much higher than 20kHz.

Yeah I agree with this recipe, except the last statement about the transformer -- use toroids and you'll easily get way more than 20kHz. 50-100kHz more likely.

Also for completeness max SPL is also determined by D/S spacing (but that is in turn determined by the LF breakpoint) and by Vpol (which is limited by the quality of your isolation).

Quote:

Another option is to only use the "small enough" section width on the first few sections of the transmission line and then increase the size of the section after that. If you look at the frequencies fed to each section you can see that as you travel down the line you pretty quickly lose the highs that would cause beaming problems. You just need to adjust the size of the transmission line resistors when you reach these sections.

I think this will worsen the polar response higher up off-axis. It's probably okay when you're more or less in the sweet spot. BTW that's how my current ESL is built

Quote:

I'll be back to work on Tuesday, and should be able to post the spreadsheet for you to experiment with then.

Thanks, could be interesting to have.

Kenneth

__________________
Never send a human to do a machine's job. --Agent Smith

Also for completeness max SPL is also determined by D/S spacing (but that is in turn determined by the LF breakpoint) and by Vpol (which is limited by the quality of your isolation).

Hello again

According to the Baxandall ESL paper and my experiments, contrary to popular belief D/S spacing does not come in to play when defining max SPL. It only places a limit on how low in frequency you can put out that max SPL due to the physical constraints of diaphragm motion.

The two parameters that define max SPL for an ESL are 1) area 2) breakdown voltage gradient for air. See section 3.3.9 of Baxandall chapter.

For a given D/S there are particular values of Vbias and Vsignal based on the breakdown voltage gradient which will produce maxSPL.
See section 3.2.9 of Baxandall chapter.

With the area of an ESL held constant, if you double the D/S you double the required Vbias and Vsignal required to reach that same value of max SPL. However, with double the D/S the voltage gradient in the airgap is the same.

Of course, that is theory....which I love!
In practice, the higher Vsignal must be generated by a transformer with twice the step-up ratio which may place limits on the bandwidth over which you can reach that SPL. Also, as you mentioned, higher voltages may be limited by imperfect stator insulation, or sharp edges and/or dust in the air gap.

Quote:

I think this will worsen the polar response higher up off-axis. It's probably okay when you're more or less in the sweet spot. BTW that's how my current ESL is built

Of course this is true. My point was that if he wanted to get some benefits of the narrow sections without using them over the whole panel, put them at the beginning of the line where the most high frequencies are.

According to the Baxandall ESL paper and my experiments, contrary to popular belief D/S spacing does not come in to play when defining max SPL. It only places a limit on how low in frequency you can put out that max SPL due to the physical constraints of diaphragm motion.

Ermm, yes you are right! I knew that, since I have read Baxandall's book chapter, but it somehow slipped my mind this morning My mistake.

This thread got me thinking. For some reason we European DIYers build mostly wire stators. Now it would be very easy to bring out every single wire as a separate signal once the wires have been glued down. If I ever build another panel I will definitely do that. This'll result in very narrow sections 3...5mm and should give superior dispersion.

When I built my current panels as a young and reckless man (that was 6 years ago) I lacked the foresight to make the stators separable, so I'm kinda stuck now to my 25mm center section surrounded by 3 big sections of about 70mm. As you would expect, it's pretty beamy but in the sweet spot it's, well, sweet!

Kenneth

__________________
Never send a human to do a machine's job. --Agent Smith

Great info guys.
Can't wait to see your spread sheet program,Steve.
After I get caught up in my things that have to get done now list,I will be able to start a new set of panels, as I have just bought all of the materials to do so.
With my method of construction ,this could very easly be applied.
My little panels are only 85mm wide and do get quite loud down to 300hz in a nearfield configuration.
I had thought about taking a few swipes with a dremel tool to segment the stators ,but I'm sure this would prove difficult to reseal the edges to stop the arcing.
A new panel build would take as much or less time to make, while not having more than needed sharp edges to contend with.
I would like try this method as I don't particularly care for the beaming effect as well, either.
So it will be interesting to compare two sets of panels of identical size and exprience the effect.
From everthing I have read that segmented stators are the way to go if done properly. jer

But, we usually think in terms of applied voltage since our amplifiers and step-up transformers are good approximations of voltages sources. Take a look at equation (5) from the paper where the Walker type equation is converted to Vsignal instead of Isignal:

P=V(signal)*V(pol)*C*sqrt(f) / (d*h*sqrt(c*r))

Now sqrt(f) is in the numerator. So, with constant voltage fed to an ESL line the response will rise +3dB/octave with increasing frequency. This implicitly takes care of the dipole roll-off for low frequencies and directivity factor for high frequencies. This rise is exactly what the R-C transmission line is use to equalize out.

Exactly! Looks like the RC line is the ideal partner for a voltage source.

In The Netherlands there's a guy who took the other approach: no segmentation but using an amplifier which acts much more like a current source. (The fact that it was a direct drive amp helped in the realization of such a behavior). Check it out here:

Here is the XL spreadsheet.
I appologize for the mixing of units, but at least I labeled them.
I tried to list all the assumptions and equations on the "Notes" & "Directions" tab, but may have missed some.
Let me know if you notice something missing or the directions don't make sense.

A few observations:
1) Native response of floor to ceiling ESL is sloped +3dB with increasing frequency
2) maxSPL increases linearly with Panel width.
2) maxSPL does not change with height of line source. (for more details see the "Notes" tab)
3) maxSPL decreases -3dB for each doubling of the listening distance.
4) maxSPL is NOT dependent on diaphragm to stator spacing, although the required Vsig & Vpol required to achieve it increase linearly with D/S.

Note that this does not mean that you can use whatever spacing you want to go as low as you want.
The spacing still must be defined to allow the required diaphragm motion for maxSPL at the LF breakpoint.

5) If you want to go low by setting the LF breakpoint low, efficiency suffers unless you increase panel width to compensate.

If you want it loud, make it a hybrid.
If you want it loud and low and don't want a hybrid, make it wide.

From what I understand, SPL is an RMS value and the sound pressure values calculated from the Walker equations are peak pressures.
So, I multiplied the pressures values coming from the Walker equation by 0.7071 before calculating SPL.

There have been measurements of mid-sized flat panels which showed amplitude deviations of ~10dB within just 1° change of listening position.
That is certainly non-practical.

Here's a plot of my 16" x 46" active area single panel ESL as a function of listening angle (polar response). I get about a 10dB drop in 5 degrees. It's a little odd to me that the higher frequencies do not drop, I'm thinking that it might be the diffraction through the perf-metal.

In Frank Verwaal papers there is a table (Part #2 p.80) with the RC filters to be used for linear frequency response. Next figures show the implementation of the idea. So, am I right, assuming:1) for different stip widths the table and individual resistors per stip (Fig.#1) is applicable.2) if the ladder structure is employed (Fig.#2) then all the resistors are equal, strip width are all the same and bolsert's spreadheet is applicable, as well as the AES article we discussed.Alex